The polynomials P_n(x) = ∫_0^1 (1 − 2t) (x − t)_n d t = ∑ A_{n,k} x^k (n = 1, 2, 3, …) appear in an asymptotic expansion for Euler’s gamma function. We investigate the properties of the coefficients A_{n,k} and show that the coefficients can be expressed in terms of the unsigned Stirling numbers of the first kind. We also show that these numbers appear in series representations for some mathematical constants, like, for instance, Euler’s constant, log(2) and ζ(3).

A NEW CLASS OF POLYNOMIALS RELATED TO THE STIRLING NUMBERS AND SERIES REPRESENTATIONS FOR SOME MATHEMATICAL CONSTANTS

Munarini E.
2021-01-01

Abstract

The polynomials P_n(x) = ∫_0^1 (1 − 2t) (x − t)_n d t = ∑ A_{n,k} x^k (n = 1, 2, 3, …) appear in an asymptotic expansion for Euler’s gamma function. We investigate the properties of the coefficients A_{n,k} and show that the coefficients can be expressed in terms of the unsigned Stirling numbers of the first kind. We also show that these numbers appear in series representations for some mathematical constants, like, for instance, Euler’s constant, log(2) and ζ(3).
2021
Stirling numbers, polynomials, log-concave, Sheffer matrix, series representations, gamma function, mathematical constants, Gregory coefficients, Cauchy numbers.
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11311/1199275
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